Problem: Simplify the following expression: $ n = \dfrac{5}{9} - \dfrac{-6x - 2}{-10x - 7} $
Solution: In order to subtract expressions, they must have a common denominator. Multiply the first expression by $\dfrac{-10x - 7}{-10x - 7}$ $ \dfrac{5}{9} \times \dfrac{-10x - 7}{-10x - 7} = \dfrac{-50x - 35}{-90x - 63} $ Multiply the second expression by $\dfrac{9}{9}$ $ \dfrac{-6x - 2}{-10x - 7} \times \dfrac{9}{9} = \dfrac{-54x - 18}{-90x - 63} $ Therefore $ n = \dfrac{-50x - 35}{-90x - 63} - \dfrac{-54x - 18}{-90x - 63} $ Now the expressions have the same denominator we can simply subtract the numerators: $n = \dfrac{-50x - 35 - (-54x - 18) }{-90x - 63} $ Distribute the negative sign: $n = \dfrac{-50x - 35 + 54x + 18}{-90x - 63}$ $n = \dfrac{4x - 17}{-90x - 63}$ Simplify the expression by dividing the numerator and denominator by -1: $n = \dfrac{-4x + 17}{90x + 63}$